Bisimulation for Fluid Stochastic Petri Nets

Abstract

We propose a novel notion of fluid bisimulation equivalence that allows one to compare and reduce the behavior of labeled fluid stochastic Petri nets (LF-SPNs) while preserving their discrete and continuous properties. The underlying stochastic model for the discrete part of the LFSPNs is a continuous time Markov chain (CTMC). The performance analysis of the continuous part of the LFSPNs is accomplished via the associated stochastic fluid models. For the fluid bisimulation on the discrete markings of two LFSPNs, we require it to be a (strong) Markovian bisimulation. On the continuous markings, for every pair of Markovian bisimilar discrete markings, the fluid flow rates of the continuous places in the first LFSPN should coincide with those of the corresponding continuous places in the second LFSPN. We prove that the resulting fluid bisimulation equivalence preserves fluid density and distribution, as well as discrete and continuous performance measures.

Logical characterization of fluid equivalences

“…• The implication ↔ f l →≡ f l is valid by Proposition 2 from [75]. Let us see that the implication is strict, i.e.…”

“…Previous works and outline of the paper. The previous results on the fluid equivalences can be found in [74,75], where we have proposed a class of LFSPNs, for which we have defined fluid trace and bisimulation equivalences and investigated their interrelations. We have shown that fluid trace equivalence preserves average potential fluid change volume for the transition sequences of every certain length.…”

“…In this paper, we investigate the behavioural relations of fluid trace and bisimulation equivalences that are useful for the comparison and reduction of the behaviour of LFSPNs with a single continuous place, since these relations preserve the functionality and performability of their discrete and continuous parts. The mentioned fluid equivalences have not been considered in the literature until they have been originally constructed by the authors in [74,75]. Those papers also contain a survey of the non-behavioural (not respecting the action names) equivalence notions that have been defined on the related models.…”

“…Moreover, R −1 does not exist if for some i (1 ≤ i ≤ n) we have R ii = 0. These difficulties are avoided in the alternative solution method for F (x), called spectral decomposition[76,54,43,49,46,74,75].…”

Behavioural equivalences for fluid stochastic Petri nets

“…The first results on this subject can be found in [73], where we have proposed a class of LFSPNs and defined a novel behavioural relation of fluid bisimulation equivalence for them. We have also proven there that the equivalence preserves aggregate fluid density and distribution, as well as discrete and continuous performance measures.…”